In ordinary Yang-Mills theory, one has two equations:
- Yang-Mills equation: $D\ast F=\ast j$, and
- Bianchi identity: $DF=0$,
where $D$ denotes the covariant derivative, $F$ is the field strength and $j$ is the electric current.
Whereas the Yang-Mills equation is something that only holds on-shell, the Bianchi identity is a purely topological/geometrical property and holds for all connections $A$ (gauge fields in physics terminology). In this way, the YM equation simply selects a subset of the full moduli space of principal connections.
In the case of the circle group $U(1)$, these equations reduce to Maxwell's equations, with the YM equation corresponding to the electric Gauss law and Maxwell's law and the Bianchi identity corresponding to the magnetic Gauss law and Faraday's law. All is fine up to here.
Now, adding a magnetic current to these equations is straightforward and actually makes the equations even more symmetric. However, doing so destroys the interpretation of $A$ as a connection 1-form and $F$ as its curvature, since the Bianchi identity won't hold anymore.
Personally, I can think of two possibilities:
- We give up on the interpretation of connections and just be pragmatic (a totally fine physical standpoint).
- We have to generalize the geometric structure.
With regards to item 2, are there other possibilities besides 2-connections on principal 2-bundles/bundle gerbes? Here, the there is also a 2-form contribution to the curvature that also contributes an extra term to the would-be Bianchi identity.
I was wondering if there are other geometric quantities that could describe this situation?
